
Then by demanding the reparameterization invariance of the point particle action, the dynamics for the compact objects is governed by the worldline point particle action given by [1]},
\(S_{\text{pp}} = \sum _{a=1,2} c \int d\tau \left( - m_{(a)} \sqrt{u_{(a)}^2} -\frac{1}{2} S_{(a)\mu \nu } \Omega _{(a)}^{\mu \nu } - \frac{S_{(a)\mu \nu }u_{(a)}^\nu }{u_{(a)}^2}\frac{du_{(a)}^\mu }{d\tau } +\mathcal {L}_{(a)\text{SI}} \right)\, ,\) 
